### Video Transcript

Given that 𝐴𝐶 is equal to 12
centimeters, 𝐵𝐶 is equal to six centimeters, and 𝐴𝐵 is equal to 13 centimeters,
what type is the largest angle?

In this question, we are given the
lengths of three sides in a triangle and asked to determine the type of the largest
internal angle in the triangle. To do this, we will compare the
side lengths to those in a right triangle.

We note that side 𝐴𝐵 is the
longest with length 13 centimeters. In a right triangle, the
Pythagorean theorem tells us that the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the two shorter sides. So 𝐴𝐵 squared is equal to 𝐴𝐶
squared plus 𝐵𝐶 squared.

We can also recall that the
Pythagorean inequality theorem allows us to determine the type of the largest angle
by comparing the square of the length of the longest side to the sum of the squares
of the lengths of the two shorter sides. In this case, if 𝐴𝐵 squared is
greater than 𝐴𝐶 squared plus 𝐵𝐶 squared, then the angle at 𝐶 is obtuse. And if 𝐴𝐵 squared is less than
𝐴𝐶 squared plus 𝐵𝐶 squared, then the angle at 𝐶 is acute.

We can calculate both of these
values for the given triangle. We have that 𝐴𝐵 squared is 13
squared, which is equal to 169. And 𝐴𝐶 squared plus 𝐵𝐶 squared
is equal to 12 squared plus six squared, which is equal to 180. We can see that the square of the
length of the longest side is less than the sum of the squares of the two shorter
sides. So the largest angle in the
triangle must be an acute angle.